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For first order kinetics, the first order effectiveness factor, t^.,, can be correlated with a modified first order Thiele modulus, <J>lnl, defined by Eq. 15.12 except for the use of the characteristic film thickness, Lk:2N

(Recall that pltl and XljM, are the same.) It should be noted that qM/Ks is equivalent to k^., the mean reaction rate coefficient defined Eq. 3.45. The correlation between and <t>lm is shown in Figure 18.12, and is described well by the first order, non-spherical form for homogeneous reaction media proposed by Aris:'

The points in the figures were computed for a variety of bioparticle and carrier particle sizes, thereby demonstrating that the characteristic film thickness works well as a normalizing factor.

Because the characteristic biofilm thickness serves as a normalizing factor that allows effectiveness factor correlations developed for planar coordinates to be used with spherical particles, it should be possible to use the general correlation for Monod kinetics shown in Figure 15.9 by using an appropriately modified Thiele modulus. This would allow external mass transfer resistance to be handled with little additional effort. Therefore, depending on the kinetic and mass transfer characteristics of the system, Eq. 18.20, Eq. 18.22, or Figure 15.9 can be used to calculate the substrate removal rate by bioparticles surrounded by substrate at a given concentration. This information can then be used to calculate system performance in the same manner as used in Chapters 15-17.

Another approach used in Chapter 15 for determining substrate removal rates by biofilms is the pseudoanalytical approach with a steady-state biofilm. The assumption of a steady-state biofilm is consistent with the situation encountered in packed towers and rotating disk reactors, but is inconsistent with FBBRs from which bioparticles are constantly wasted, as discussed previously. Thus, while FBBR models are available that assume steady-state biofilms, one must question their relevance to most operating FBBRs. As a consequence, steady-state biofilm FBBR models are not discussed here.

### 18.3.2 Fluidization Submodel

The effect of the fluidization regime on the thickness of biofilm that can be maintained on carrier particles of a given size and density constrained within a bed of fixed height was discussed in Section 18.2.3. Figure 18.9, shown there, presents the algorithm for calculating that biofilm thickness, and as such, represents a fluidization submodel that can be used. No further discussion of it is needed here. It should be noted, however, that such a model assumes a uniform biofilm thickness throughout the FBBR, which may not conform to reality for some FBBRs, as pointed out previously. Other models'21 are capable of handling variations in particle size within

Figure 18.12 Relationship between the bioparticle first-order effectiveness factor, t|c1, and the modified firstorder Theile modulus, tj>lm. (From W. K. Shieh and J. D. Keenan, Fluidized bed biofilm reactor for wastewater treatment. Advances in Biochemical Engineering/Biotechnology 33:131-169, 1986. Copyright © SpringerVerlag New York, Inc.; reprinted with permission.)

Figure 18.12 Relationship between the bioparticle first-order effectiveness factor, t|c1, and the modified firstorder Theile modulus, tj>lm. (From W. K. Shieh and J. D. Keenan, Fluidized bed biofilm reactor for wastewater treatment. Advances in Biochemical Engineering/Biotechnology 33:131-169, 1986. Copyright © SpringerVerlag New York, Inc.; reprinted with permission.)

the bed, but because of space constraints and their added complexity, they are not discussed here. Rather, the reader is referred to the cited papers.

### 18.3.3 Reactor Flow Submodel

The reactor flow submodel must link the biofilm and fluidization submodels to allow computation of the performance of an FBBR. The type of model that should be used depends on the hydraulic regime in the FBBR. The nature of that regime is determined primarily by the degree of substrate utilization across the bed and the amount of effluent recirculated to maintain the appropriate fluidization velocity and to provide the required amount of electron acceptor. If the recirculation ratio is high and the influent substrate concentration is low, so that the change in substrate concentration across the bed (after dilution of the influent by the recirculation flow) is small, then the bed can be considered to behave as if it were a completely mixed reactor. The validity of this approach can be checked easily by comparing the diluted influent concentration as calculated with Eq. 16.5 to the assumed effluent concentration. On the other hand, if the recirculation ratio is low (e.g., <2) or the degree of dilution is small, then the FBBR must be treated as a plug—flow reactor, either with or without axial dispersion, or as a series of continuous stirred tank reactors (CSTRs). However, consideration need not be given to changes in the degree of axial dispersion from point to point in an FBBR due to differences in the porosity because that level of complexity cannot be justified.1 Examples of all of these approaches can be found in the literature, depending on the situation being modeled. The equations used are typical of these various flow regimes as discussed in previous chapters and thus are not presented here.

Basically the approach to FBBR modeling is iterative, with the number of loops depending on the reactor flow submodel. Only a completely mixed FBBR (both bioparticles and liquid) is considered to describe the procedure, but the concepts can be extended to other flow regimes, which usually require more iterative loops. First, the characteristics of the FBBR must be established, including the desired superficial velocity, v, the FBBR cross-sectional area, A^, the desired fluidized bed height, H,,,,, the mass of carrier particles, Mp, their diameter, dp, and their density, pp. The biofilm thickness that can be maintained by these conditions can then be computed using the procedure in Figure 18.9. That thickness determines the bioparticle diameter, which defines the characteristic biofilm thickness, Lu, which is used to determine the effectiveness factor in the biofilm submodel. If the effectiveness factor expression includes the bulk substrate concentration, then one must be assumed. It is equivalent to the effluent substrate concentration for a completely mixed FBBR. The biofilm submodel is then used in the reactor flow submodel to compute the output substrate concentration. This is a direct computation for a completely mixed FBBR, but an iterative procedure is required for a plug-flow or tanks-in-series flow regime. If the computed concentration is different from the assumed value, then a new value must be assumed and the procedure repeated until the computed effluent concentration agrees with the value assumed. This is the effluent substrate concentration from the FBBR. The entire procedure can be repeated for different initial conditions, i.e., v, Mp, dp, or pp, thereby relating performance to those conditions. This allows identification of the conditions giving an effluent concentration equal to or less than some desired value.

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